Actions of small cancellation groups on hyperbolic spaces
We generalize Gruber–Sisto’s construction of the coned-off graph of a small cancellation group to build a partially ordered set TC of cobounded actions of a given small cancellation group whose smallest element is the action on the Gruber–Sisto coned-off graph. In almost all cases TC is incredibly r...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Springer Verlag
2020
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| _version_ | 1826286808064851968 |
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| author | Abbott, C Hume, D |
| author_facet | Abbott, C Hume, D |
| author_sort | Abbott, C |
| collection | OXFORD |
| description | We generalize Gruber–Sisto’s construction of the coned-off graph of a small cancellation group to build a partially ordered set TC of cobounded actions of a given small cancellation group whose smallest element is the action on the Gruber–Sisto coned-off graph. In almost all cases TC is incredibly rich: it has a largest element if and only if it has exactly 1 element, and given any two distinct comparable actions [G↷X]⪯[G↷Y] in this poset, there is an embeddeding ι:P(ω)→TC such that ι(∅)=[G↷X] and ι(N)=[G↷Y]. We use this poset to prove that there are uncountably many quasi-isometry classes of finitely generated group which admit two cobounded acylindrical actions on hyperbolic spaces such that there is no action on a hyperbolic space which is larger than both. |
| first_indexed | 2024-03-07T01:49:13Z |
| format | Journal article |
| id | oxford-uuid:997d1e99-d621-4b84-b323-65e93ff1d816 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T01:49:13Z |
| publishDate | 2020 |
| publisher | Springer Verlag |
| record_format | dspace |
| spelling | oxford-uuid:997d1e99-d621-4b84-b323-65e93ff1d8162022-03-27T00:14:43ZActions of small cancellation groups on hyperbolic spacesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:997d1e99-d621-4b84-b323-65e93ff1d816EnglishSymplectic ElementsSpringer Verlag2020Abbott, CHume, DWe generalize Gruber–Sisto’s construction of the coned-off graph of a small cancellation group to build a partially ordered set TC of cobounded actions of a given small cancellation group whose smallest element is the action on the Gruber–Sisto coned-off graph. In almost all cases TC is incredibly rich: it has a largest element if and only if it has exactly 1 element, and given any two distinct comparable actions [G↷X]⪯[G↷Y] in this poset, there is an embeddeding ι:P(ω)→TC such that ι(∅)=[G↷X] and ι(N)=[G↷Y]. We use this poset to prove that there are uncountably many quasi-isometry classes of finitely generated group which admit two cobounded acylindrical actions on hyperbolic spaces such that there is no action on a hyperbolic space which is larger than both. |
| spellingShingle | Abbott, C Hume, D Actions of small cancellation groups on hyperbolic spaces |
| title | Actions of small cancellation groups on hyperbolic spaces |
| title_full | Actions of small cancellation groups on hyperbolic spaces |
| title_fullStr | Actions of small cancellation groups on hyperbolic spaces |
| title_full_unstemmed | Actions of small cancellation groups on hyperbolic spaces |
| title_short | Actions of small cancellation groups on hyperbolic spaces |
| title_sort | actions of small cancellation groups on hyperbolic spaces |
| work_keys_str_mv | AT abbottc actionsofsmallcancellationgroupsonhyperbolicspaces AT humed actionsofsmallcancellationgroupsonhyperbolicspaces |